Finite Group Databasebeta

← #309101

Subgroups — #309101

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28814C211yesmaximal, characteristic, complemented
2#26532C211yesmaximal, characteristic, complemented
3#26526C211yesmaximal, characteristic
4#23216C211yesmaximal, characteristic
5#10596C211yesmaximal, characteristic, complemented
6#10595C211yesmaximal, characteristic
7#10357C211yesmaximal, characteristic, complemented
8#10354C211yesmaximal, characteristic
9#9830C211yesmaximal, characteristic
10#9738C211yesmaximal, characteristic
11#9614C211yesmaximal, characteristic, complemented
12#9598C211yesmaximal, characteristic, complemented
13#8926C211yesmaximal, characteristic, complemented
14#8921C211yesmaximal, characteristic
15#8284C211yesmaximal, characteristic, complemented
16C2×#499—21no—
17C2×#486C2211yescharacteristic
18#2552C2211yescharacteristic
19#2552C2211yescharacteristic
20#2366C2211yescharacteristic
21#2365C2211yescharacteristic
22#2095—21no—
23#2093—21no—
24#2068—21no—
25#2067—21no—
26#2059—21no—
27#2059—21no—
28#1990C2211yescharacteristic
29#1988C2211yescharacteristic
30#1982C2211yescharacteristic
31#1982C2211yescharacteristic
32#1981C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1942—21no—
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1934—21no—
44#1925—21no—
45#1863—21no—
46#1862—21no—
47#1862—21no—
48#1845C2211yescharacteristic
49#1803—21no—
50#1802—21no—
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#499—44no—
75#486—22no—
76#486Q814yes—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107D411yescharacteristic
82C2×#107D411yescharacteristic
83C2×#107D411yescharacteristic
84C2×#107D411yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—41no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#398—41no—
112#398—41no—
113#391—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#387—41no—
122#386—41no—
123#384—21no—
124#384—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—41no—
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#335—22no—
148#334—22no—
149#332—22no—
150#331—22no—
151#322—22no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C23×C4—41no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C2×Q812yes—
170#127—21no—
171#127—21no—
172#127—21no—
173#127C2×Q812yes—
174#125—42no—
175#125—42no—
176#125—81no—
177#125—81no—
178C4×Q8—42no—
179C4×Q8—81no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—81no—
183C4×D4—81no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—41no—
215C2×C42C2411yescharacteristic
216#107—22no—
217#107—22no—
218#107—22no—
219#107—22no—
220#107—42no—
221#107—81no—
222#107—81no—
223#107—81no—
224#106—42no—
225#106—42no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8#4112yes—
236#103—42no—
237#103—81no—
238#103—81no—
239#103—81no—
240#102—24no—
241#102—24no—
242#102—42no—
243#102—42no—
244#102—81no—
245#102—81no—
246#102—81no—
247#102—81no—
248#102—82no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—42no—
254C4×C8—81no—
255C4×C8C2×Q812yes—
256C4×C8#4112yes—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265C24—41no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—81no—
287C2×D4—81no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—161no—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8C22⋊Q812yes—
330C2×C8C22⋊Q812yes—
331C2×C8#13512yes—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—161no—
365C4⋊C4—161no—
366C4⋊C4#13512yes—
367C4⋊C4AGL(1,ℤ/8)12yes—
368C4⋊C4#13712yes—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—161no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—82no—
381C42C22×Q811yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C422−1+411yescharacteristic
385C23—42no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23D4211yescharacteristic
393Q8—42no—
394Q8—42no—
395Q8—81no—
396Q8—81no—
397D4—42no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—82no—
402D4—81no—
403D4—81no—
404D4—81no—
405D4—81no—
406D4—161no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—164no—
443C2×C4C2×C22⋊Q811yescharacteristic
444C2×C4#53211yescharacteristic
445C2×C4#54211yescharacteristic
446C2×C4#54811yescharacteristic
447C2×C4D4×Q811yescharacteristic
448C2×C4#55411yescharacteristic
449C2×C4C2×#13511yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C2×C4#57611yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57811yescharacteristic
457C8—22no—
458C8—22no—
459C8—22no—
460C8—42no—
461C8—41no—
462C8—81no—
463C8—81no—
464C8—81no—
465C8—81no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22#245211yescharacteristic
474C22#326411yescharacteristic
475C22#327011yescharacteristic
476C22#328011yescharacteristic
477C22#328711yescharacteristic
478C22#328811yescharacteristic
479C22#328911yescharacteristic
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—82no—
489C4—81no—
490C4—81no—
491C4—161no—
492C4—161no—
493C4#307911yescharacteristic
494C4#308311yescharacteristic
495C4#316811yescharacteristic
496C4#317611yescharacteristic
497C4#336711yescharacteristic
498C4#337511yescharacteristic
499C4#338011yescharacteristic
500C4#338011yescharacteristic
501C2—162no—
502C2#1977211yescharacteristic
503C2#1978111yescharacteristic
504C2#1986711yescharacteristic
505C2#1987311yescharacteristic
506C2#2647811yescharacteristic
507C2#2648211yescharacteristic
508C2#2650311yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #28814C2semidirect
2. #26532C2semidirect
5. #10596C2semidirect
7. #10357C2semidirect
11. #9614C2semidirect
12. #9598C2semidirect
13. #8926C2semidirect
15. #8284C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#288142
#265322
#265262
#232162
#105962
#105952
#103572
#103542
#98302
#97382
#96142
#95982
#89262
#89212
#82842